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extracta mathematicae Vol. 18, N´um. 3, 315 – 319 (2003)
On Banach Spaces Containing Complemented and Uncomplemented Subspaces Isomorphic to c0 ´ i Medina Galego, Anatolij Plichko Elo Department of Mathematics - IME, University of S˜ ao Paulo, S˜ ao Paulo 05315-970, Brazil Politechnika Krakowska im. Tadeusza Ko´sciuszki, Instytut Matematyki, ul. Warszawska 24 Krak´ ow 31-155 Poland e-mail:
[email protected],
[email protected] (Presented by J.M.F. Castillo)
AMS Subject Class. (2000): 46B26, 46B25
Received December 8, 2003
1. Introduction In this short note we give a negative answer to a question of Argyros, Castillo, Granero, Jim´enez and Moreno concerning to Banach spaces which contain complemented and uncomplemented subspaces isomorphic to c0 . To be more precise, let X be a Banach space, following [1] we say that X is separably Sobczyk if every subspace of X isomorphic to c0 is complemented in X. We also say that X is hereditarily separably Sobczyk space if every subspace Y of X which is isomorphic to c0 , contains a subspace Z isomorphic to c0 and complemented in X. In [1, page 754] was posed the following question: Question 1.1. Suppose that a Banach space X is hereditarily separably Sobczyk. Does it follow that X is separably Sobczyk space? Next we will present several examples of Banach spaces which give negative answers to this question. The first one that we obtained is constructed from some n-Sobczyk spaces introduced in [9]. Then, we realized that even for C(K) spaces, where K is dispersed compact, the answer to above question is negative. Finally, we show that every Banach space which contains no subspace isomorphic to l1 is hereditarily separably Sobczyk. In particular, the well known Johnson-Lindenstraus space JL [6, Example 1, page 222] is also a counterexample to Question 1.1. 315
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Let us recall that a Banach space is n-Sobczyk if every K-isomorphic copy of c0 therein is the range of a projection with norm at most nK. In [9] was introduced examples of Banach spaces X for which inf{λ : X is λ-Sobczyk} is arbitrarily large. Moreover, each one of these spaces X contains a subspace isometric to c0 . In order to present a solution to Question 1.1, we fix by this previous result, a sequence (Xn )n∈N of separably Sobczyk spaces having the following property: (1) There is a subspace Xn0 of Xn isometric to c0 , such that every projection of Xn onto Xn0 has norm greater or equal than n. P Theorem 2.1. Let X = ( ∞ 1 Xn )c0 be the c0 -sum of (Xn )n∈N . Then (a) X is hereditarily separably Sobczyk. (b) X is not separably Sobczyk. Proof. (a) Assume that Y is a subspace of X isomorphic to c0 . Denote by dn the distance dist(S(Y ), [(Xi )∞ i=n ]) between the unit sphere S(Y ) of Y and ∞ the closed subspace [(Xi )i=n ] of X generated by (Xi )∞ i=n . We distinguish two possible cases: First case: dn = 0 for every n ∈ N. Therefore for 0 < ε < 1 there exist a strictly increasing sequence (nk )k∈N nk+1 in N and sequences (yk )k∈N in S(Y ) and (zk )k∈N in S([(Xi )i=n ]) with k +1 (2)
kyk − zk k < ε2−k , k = 1, 2, . . .
Obviously, (zk )k∈N is equivalent to the unit vector basis of c0 and there is a projection P : X → [(zk )k∈N ] with kP k = 1 . So by (2) we deduce that dist(S([(yk )k∈N ]), kerP ) > 1 − ε . Since [(yk )k∈N ] ⊕ kerP = X, it follows that there is a projection Q : X → [(yk )k∈N ] with kQk < (1 − ε)−1 . By (2), we conclude that the subspace [yk ]∞ 1 is isomorphic to c0 . Second case: dn+1 > 0 for some n ∈ N. In this case, the natural projection Qn : X → [(Xk )nk=1 ], restricted to Y , is an isomorphism onto the subspace Qn Y which is isomorphic to c0 . Moreover, it easy to see by (1) that there is a bounded projection R : [Xk ]nk=1 → Qn Y. Thus Q−1 n RQn is also a bounded projection of X onto Y .
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¡ P∞ 0 ¢ 0 0 (b) Let X0 = 1 Xn c0 be the c0 -sum of (Xn )n∈N , where Xn is the Banach space mentioned in (1). Clearly X0 is isometric to c0 . Now suppose that there exists a bounded projection P : X → X0 and take m ∈ N satisfying kP k < m. Thus for the natural projection Pn : X → Xn the operator Pn P |Xn would be a projection of Xn onto Xn0 with the norm less or equal than m, which gives a contradiction for n > m. 3. An example from the C(K) spaces In [7, Theorem 11] Lotz, Peck and Porta proved that a compact space K is scattered if and only if every infinite dimensional subspace of C(K) contains a subspace isomorphic to c0 and complemented in C(K). Therefore, in the case where K is a dispersed compact, C(K) is hereditarily separably Sobczyk space. However Molt´o [8] has constructed a scattered compact K such that C(K) has a subspace isometric to c0 which is not complemented in C(K). Thus, this C(K) is not separably Sobczyk space. 4. Some more hereditarily separably Sobczyk spaces In [2, Theorem 2.1] was observed that the following reformulation of a result of Hagler and Johnson [5, Theorem 1.a] is true. Theorem 4.1. Let X be a real Banach space and (x∗n )n∈N a sequence in X ∗ equivalent to the unit vector basis of l1 . If no normalized l1 -block of (x∗n )n∈N is weak∗ null sequence, then X contains a subspace isomorphic to l1 . As a consequence of this reformulation, Diaz and Fern´ andez proved: Theorem 4.2. Let X be a real Banach space that does not contain a subspace isomorphic of l1 . If X contains a subspace isomorphic to c0 , then X contains a complemented subspace Z isomorphic to c0 . Moreover, for every ² > 0, we can find the subspace Z so that there is a projection P : X → Z with kP k < 1 + ². In [4, Proposition 2.1] was noted that one can slightly modify the proof of Theorem 4.2 to show that every Banach space which contains no subspace isomorphic to l1 is hereditarily separably Sobczyk space. Since no proof was presented in [4], for sake of completeness, we will prove Theorem 4.3.
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Theorem 4.3. Let X be a real Banach space that does not contain a subspace isomorphic to l1 . If Y is a closed subspace of X that contains a subspace isomorphic to c0 , then Y contains a subspace Z isomorphic to c0 which is complemented in X. Moreover, for every ² > 0, we can find the subspace Z so that there is a projection P : X → Z with kP k < 1 + ². Proof. Fix ε > 0. Since Y contains a subspace isomorphic to c0 , by James distortion theorem [3, Theorem 1, XIV], taking δ = ²/(1 + ²), there is a sequence (yn )n∈N in the unit ball of Y such that n °X ° ° ° (3) (1 − δ) sup |ak | ≤ ° ak yk ° ≤ sup |ak | k
k
k=1
for all scalars (ak )nk=1 and all n ∈ N. ∗ by For each m ∈ N define ym ∗ ym
n ³X
´ a k yk = a m .
k=1 ∗ to an element of X ∗ , that we By the Hahn-Banach theorem we can extend ym ∗ still denote ym , so that
1 = 1 + ², 1−δ for all m, n = 1, 2, . . ., where δmn denotes the Kronecker delta. The sequence (yn∗ )n∈N is equivalent to the unit vector basis of l1 in X ∗ . In fact, we have that n n n ° °X X 1 X ° ° |ak |, ak yk∗ ° ≤ |ak | ≤ ° 1−δ ∗ ∗ ym (yn ) = δmn and kym k≤
(4)
k=1
k=1
k=1
for all scalars (ak )nk=1 and for all n ∈ N. Notice that if a1 , a2 , . . . , an are real scalars, we can consider numbers ²k with ²k ak = |ak |, k = 1, 2, . . .. Then n n n n n ¯ X ¯ ¯X °X ° ¯X X ¯ ¯ ¯ ¯ ° ∗° ∗ ²k ak ¯ = |ak |. ²k xk )¯ = ¯ ak xk ° ≥ ¯ ak xk ( ° k=1
k=1
k=1
k=1 ∗
k=1
According to Theorem 4.1, there is a weak null sequence (zn∗ )n∈N which is a normalized l1 -block of (yn∗ )n∈N . It has the following form X zn∗ = ak yk∗ , k∈An
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where(A n )n∈N is a sequence of pairwise disjoint finite subsets of N, and P |a k∈An k | = 1, for every n ∈ N. Put X ²k yk , zn = k∈An
where ²k = signak , for all k ∈ An and n ∈ N. Thus, we have constructed sequences (zn )n∈N in Y and (zn∗ )n∈N in X ∗ such that (5) (3) remains unchanged for (zn )n∈N . (6) (zn∗ )n∈N is a weak∗ null sequence in X ∗ that is equivalent to the unit vector basis of l1 , and (4) remains unchanged for (zn∗ )n∈N . Let us consider the mapping P from X onto the closed subspace Z generated by (zn )n∈N ∞ X P (x) = zn∗ (x)zn . 1
Now from (5) and (6), it is easy to verify that Z is isomorphic to c0 and that P is the projection from X onto Z with kP k ≤
1 =1+². 1−δ
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´ ndez, A., Reflexivity in Banach lattices, Arch. Math. 63 [2] Diaz, S., Ferna (1994), 549 – 552.
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[4] Galego, E.M., On subspaces and quotients of Banach spaces C(K, X), Monatsh. Math. 136, No.2 (2002), 87 – 97.
[5] Hagler, J., Johnson, W.B., On Banach spaces whose dual balls are not [6] [7] [8] [9]
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